$L^p$ estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have $k$th order contact

In this paper we develop $L^p$ estimates for functions $u$ which are joint quasimodes of semiclassical pseudodifferential operators $p_1(x,hD)$ and $p_2(x,hD)$ whose characteristic sets meet with $k$th order contact, $k\ge 1$. As part of the technical development we use Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of $p_1(x,\xi )$.